Continuous Charge Distribution

We have dealt with the systems containing discrete charges such as Q1, Q2,…,Qn. We noticed that the operations involved in such cases were simpler and rules of scalar addition could be easily applied. But as we know, at times we need to deal with systems where considering the charges discrete is not an option. Here, we need to consider the charge as those which are continuously distributed over a length or a surface or a volume.

For example, if we are dealing with a surface carrying a continuous charge distribution in the body over its surface, we cannot calculate the value of electric field due to each microscopic charged constituent. We rather consider an area element, very small in size, but big enough to include a large number of such charged constituents which we can denote as Δs.

Let us say the total charge carried by this area element is equal to ΔQ, and then the charge density of the element can be given as,

The unit of σ is C/m2 or coulomb per square metres.

Similarly, in case of charge distribution in a body along a line segment of length Δl such as a small line element of wire, the linear charge density can be given as,

Where ΔQ is the charge contained in that line element. The unit of λ is C/m or coulomb per metre.

Similarly, when the charge ΔQ is distributed over a microscopically small volume element ΔV, the volume charge density can be given as,

The unit of ρ is C/m3or coulomb per cubic metres.

Calculation of electric field

Let us consider a case of the continuous charge distribution in a body. Here, we will calculate the electric field due to this charge at a point P. We can say that the charge density at different volumetric elements can be different, so we divide the body into different elements such that the charge density for a particular element can be considered to be a fixed quantity. Consider one such element of volume Δv, whose charge density is given by ρ. Let the distance of the volume element from the point P be given as r. The charge in the volume element can be given as ρΔv. As per the Coulomb’s law, the electric field due to the charge ρΔv can be given as,

Here, r is the distance between the charged element and the point P at which the field is to be calculated and ř is the unit vector in the direction of the electric field from the charge to the point P.

By the principle of superposition, the electric field due to the total charge distribution in a body, distributed into several such volume elements, can be given as,